lecture09

lecture09 - Yinyu Ye, MS&E, Stanford...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 1 ApplicationsofKKTConditionsandNonlinearOptimization YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 2 PortfolioManagement Let r denotethe expectedreturnvector and V denotethe co-variancematrix of aninvestmentportfolio,andlet x betheinvestmentproportionvector. minimize x T V x subjectto r T x , e T x =1 , x , where e isthevectorofallones.Thisisa quadraticprogram . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 3 AllowShortOption minimize x T V x subjectto r T x , e T x =1 . OptimalityConditions: 2 V x- r- e = , , ( r T x- )=0 . =2 x T V x- . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 4 Solutiontolinearequations Trycase r T x > ,thatis, =0 : 2 V- e e T x = 1 andverifywhetherornot r T x > ;ifnotsolve 2 V- r- e r T e T x = 1 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 5 PortfolioExample r 1 =0 . 0825 ,r 2 =0 . 1125 , V = . 01390 . 0056 . 00560 . 0261 =0 . 1 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 6 KnapsackMaximization maximize j f j ( x j ) subjectto j p j x j w, x j , j OptimalityConditions: - f j ( x j ) p j , ,x j - f j ( x j ) p j =0 , ( w- X j p j x j )=0 , j. Implications: = f j ( x j ) p j , x j > 0; and f j ( x j ) p j , x j =0 Investonlytotheprojectswhose marginalreturnperdollar isaboveathreshold . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #09 7 FishersExchangeMarket Buyers havemoney( w i...
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lecture09 - Yinyu Ye, MS&E, Stanford...

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